The generalization of the Killing spinor equation to nonconstant Killing function $\lambda$ has been worked out in
H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).
A nonconstant $\lambda$ is only possible if its real part is identically zero.
A more recent paper along these lines is Complex Generalized Killing Spinors on Riemannian Spin manifolds (2013).